The realization space is
  [1   0   1                  1   0   1    0    1                  1    1               1]
  [0   1   1   x1^3 - x1^2 + x1   0   0    1    1   x1^3 - x1^2 + x1   x1              x1]
  [0   0   0                  0   1   1   -1   x1      x1^2 - x1 + 1    1   x1^2 - x1 + 1]
in the multivariate polynomial ring in 1 variable over ZZ
within the vanishing set of the ideal
Ideal (x1^13 - 4*x1^12 + 8*x1^11 - 10*x1^10 + 7*x1^9 - 7*x1^7 + 10*x1^6 - 8*x1^5 + 4*x1^4 - x1^3)
avoiding the zero loci of the polynomials
RingElem[x1, x1 - 1, x1^2 - x1 + 1, x1^4 - 2*x1^3 + 2*x1^2 - x1 + 1, x1^2 + 1, x1^3 - 2*x1^2 + x1 - 1, x1 + 1, x1^3 - x1^2 - 1]